I would go with the first sentence.
The current machine has been repaired.
Instead of using the definite article, the, the indefinite article, a, can also be used when the current machine is not specified. (There are many current machines.)
A current machine has been repaired.
The second sentence shouldn't be used unless Current Machine is a proper noun.
Current Machine has been repaired.
Note: Some people may consider the word "current" serves a determining function and thus not use any determiner. Grammatically speaking, it is wrong to do that. The word "current" is always classed as an adjective and thus it requires another determiner to complete the phrase.
An articles is a common kind of determiner. A noun phrase normally comes with a determiner to express its reference in the context. The determiner can be my (a possessive), John's (a nominal possessive), two (a cardinal number), this (a demonstrative), etc. No determiner should be used when the noun is a pronoun or proper noun, or another determiner has already been used.
Note: Sometimes it is not wrong to use a determiner even when it is a proper noun or pronoun: "My Victoria is on her way to my house." "She was the one who taught me Spanish."
There are scenarios when no determiner is used even though it is not a pronoun or proper noun. It is a type of zero-marking, namely zero-article. No determiner (or article) should be used in front of a mass noun or a plural noun that is generic or indefinite in the sentence.
generic mass noun: Happiness is contagious.
generic plural noun: Cars have accelerators.
indefinite mass noun: I drink coffee.
indefinite plural noun: Friends have told us that they like our new house.
Note: Article is sometimes omitted before some words for specific institutions, such as school: "She's going to school." Article may also be omitted between a preposition and the word bed when describing activities typically associated with beds: "She went to bed." But when describing activities that are not typical, or a particular location is meant, the definite article is used: "I went to the school to see my son." "We were jumping on the bed."